[dragon96]

The physics I've done have only done with n-particle systems, where n is a number I can count on a hand. Likewise, most of the examples I'd learn in a pure math class are either (a) relatively well-behaved objects with nice properties or (b) "pathological" objects that are constructed specifically to prove a counterpoint.

The kind of pathological function that I'm referring to is neither of these. For example, what does the manifold of all 1 second clips of the word "the" look like? If the clip is sampled at 60 Hz, each clip is already in 60d space. I'm inclined to think that it's some unimaginably complicated manifold that would likely fall into the category of "edge cases", which previous commenters have mentioned and it sounds like you're discounting as nitpicking.

I don't know if this aligns with what you mean by a pathological function, and I'm happy to continue having this discussion with a more concrete example of what you mean. :)